c Dr. Craig Gin
Math 141, Fall 2015, Math 141 Week-in-Review 6
Sections 6.4 and 7.1 and Review for Exam 2
Key Terms:
• factorial: For any natural number n, n! = n(n − 1)(n − 2) · · · 3 · 2 · 1
• permutation: a permutation of a set of elements is an ordered arrangement of all the
elements.
• experiment: an activity with observable results.
• outcome: the result of an experiment.
• sample space: the set of all outcomes of an experiment.
• event: a subset of the sample space. An event E is said to occur whenever E contains
the observed outcome.
• mutually exclusive: Two events E and F are said to be mutually exclusive if E ∩F = ∅.
Relevant Equations:
• The number of permutations of n distinct objects is given by P (n, n) = n!
• The number of permutations of r distinct objects taken from a set of size n is
n!
given by P (n, r) = (n−r)!
• Distinguishable Permutations: Given a set of n objects in which n1 objects are alike
and of one kind , n2 objects are alike and of another kind, ..., and nm objects are alike and
of yet another kind, so that n1 + n2 + ... + nm = n then the number of distinguishable
permutations of these n objects taken n at a time is given by n1 !n2n!!···nm ! .
• The number of combinations of r distinct objects taken from a set of size n,
n!
denoted by C(n, r), is given by C(n, r) = (n−r)!r!
Connections between set theory and probability terminology:
• sample space ⇐⇒ universal set
• event ⇐⇒ set
• outcome ⇐⇒ element
• impossible event ⇐⇒ empty set
• certain event ⇐⇒ universal set
• mutually exclusive ⇐⇒ disjoint
• Note: Union, intersection, and complement are all the same
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c Dr. Craig Gin
Math 141, Fall 2015, 1. A group of 12 people contains 7 males and 5 females. A photographer wants a picture
of 7 of them: 4 men in the back row and 3 women in the front row. How many different
arrangements can be made?
2. A convenience store has three different flavors of beef jerky. There are 8 bags of
mild, 6 bags of hot, and 5 bags of teriyaki. If two bags of the same flavor can’t be
distinguished, in how many distiguishable ways can the bags be arranged in a single
row on the shelf?
3. A bag of M&M’s contains 6 blue M&M’s, 5 red M&M’s, and 3 yellow M&M’s.
(a) In how many ways can a sample of 5 M&M’s be selected from the bag?
(b) In how many ways can such a sample consist of exactly 2 red M&M’s?
(c) In how many ways can the sample consist of at least 3 blue M&M’s?
(d) In how many ways can the sample consist of exactly 2 red or exactly 3 blue
M&M’s?
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c Dr. Craig Gin
Math 141, Fall 2015, 4. Adam has a collection of 10 DVDs. 5 of them are action movies, 3 are comedy, and
2 are drama. In how many ways can Adam arrange the 10 DVDs if each genre must
stay together?
5. A congressional committee needs to be formed that includes 6 members. One of the
members will be designated the chair and one the vice-chair. If this committee must
be formed from a pool of 20 representatives, how many different committees can be
formed?
6. An experiment consists of tossing a coin, rolling a die, and observing the outcomes.
(a) Determine the sample space for this experiment.
(b) Determine the event E that a head is tossed and an odd number is rolled.
(c) Determine the event F that an even number is rolled.
(d) Find E ∪ F .
(e) Are these two events mutually exclusive?
(f) How many events does this experiment have?
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c Dr. Craig Gin
Math 141, Fall 2015, Section 3.1: Graphing systems of inequalities
7. Determine graphically the solution set for the following system of inequalities. Indicate
whether it is bounded or unbounded.
3x − 2y > 6
x + 2y ≥ 14
x≤8
y
x
Section 3.2: Setting up linear programming problems
8. A financier has earmarked at most $100,000 to invest in two projects. She estimates
that project A will yield a return of 10% and project B will yield a return of 8%.
Because project A has more risk than project B, she has decided to invest at least
twice as much money in project B as she does in project A. How much should she
invest in each project to maximize the total return on her investments? Set up the
problem but do not solve.
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c Dr. Craig Gin
Math 141, Fall 2015, Section 3.3: Solving linear programming problems (method of corners)
9. Solve the following linear programming problem.
Maximize P = 5x + 3y
Subject to x − 5y ≤ −5
x + 3y ≤ 15
x+y ≤7
x ≥ 0, y ≥ 0
y
x
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c Dr. Craig Gin
Math 141, Fall 2015, Section 6.1: Sets and set operations
10. Consider the universal set U = {1, 2, 3, 4, a, b, c, d, e, f } and the subsets A = {1, 2, 3, 4},
B = {x|x is a number that is even or a letter that is a vowel}, and C = {a, b, d, e}.
(a) TRUE
FALSE
2∈A
(b) TRUE
FALSE
{a, b} ∈ C
(c) TRUE
FALSE
{1, 4} ⊂ A
(d) Write the set B in roster notation.
(e) Find A ∩ B.
(f) Find (A ∪ B) ∩ C C .
(g) Are A and C disjoint?
11. Shade the appropriate regions in the following Venn Diagrams.
(a) A ∪ B C ∪ C C
U
A
B
C
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c Dr. Craig Gin
Math 141, Fall 2015, (b) (A ∪ C C ) ∩ B
U
A
B
C
Section 6.2: Number of elements in a set
12. A survey asked 130 people what flavors of Blue Bell ice cream they like.
Let V be the set of people who like Homemade Vanilla.
Let C be the set of people who like Cookies and Cream.
Let M be the set of people who like Mint Chocolate Chip.
Use the survey results below to fill in the following Venn Diagram.
• 75 people like Homemade Vanilla.
• 89 people like Cookies and Cream.
• 93 people like at least two of the three flavors.
• 49 people like Homemade Vanilla and Mint Chocolate Chip
• 56 people like Mint Chocolate Chip and Cookies and Cream.
• 11 people only like Mint Chocolate Chip.
• 31 people like all three flavors.
U
V
C
M
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c Dr. Craig Gin
Math 141, Fall 2015, (a) How many people don’t like any of the flavors?
(b) How many people like exactly one flavor?
Section 6.3: Multiplication principle
13. A license plate consists of three letters followed by three numbers (0-9). How many
different license plates can be formed that start with an R and do not repeat letters
or numbers?
14. A group of five males and four females go to the movies. In how many ways can they
sit in a single row of nine seats if they must alternate gender?
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